Module Focus: Algebra II – Module 3
Sequence of Sessions
Overarching Objectives of this December 2014 Network Team Institute

Participants will be able to identify, practice, and use best instructional moves and scaffolds for chosen common core standards
High-Level Purpose of this Session



Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within this
module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.
Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work
of the grade in order to fully implement the curriculum.
Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while
maintaining the balance of rigor that is built into the curriculum.
Related Learning Experiences
 This session is part of a sequence of Module Focus sessions examining the Algebra II modules within the curriculum, A Story of Functions.
Key Points
●
Topic A
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Topic B
●
●
●
●
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Logarithmic functions have an important historical context.
Logarithms help us make sense of very large and very small numbers.
Logarithms give us a way of solving exponential equations.
Properties of logarithms are essential for evaluating logarithms and solving exponential or logarithmic equations.
Topic C
●
●
●
●
The domain of an exponential function is the set of all real numbers.
Therefore, exponential expressions can be evaluated for any rational or irrational value.
It is useful to be able to manipulate expressions containing exponents and/or radicals into forms that are convenient to use.
The number 𝑒 is an irrational number that is important in a variety of mathematical applications.
A logarithmic function is the inverse of an exponential function.
Properties of logarithms and exponents are useful for graphing transformations of logarithmic and exponential functions.
Using the change of base formula, any logarithmic function can be rewritten as a vertical scaling of the natural logarithm function (or any other base
logarithm).
Topic D
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●
●
●
When modeling with exponential functions, logarithms are useful for solving the exponential equations that arise.
The number e appears in many applications of exponential functions.
There is a powerful connection between geometric sequences, exponential functions, and logarithms.
We will continue to explore applications of exponentials and logarithms in Topic E.
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Topic E
●
●
●
●
Geometric sequences and series are prevalent in finance calculations.
Good modeling problems are often open-ended and sometimes messy!
Technology is useful for exploring a problem graphically or numerically.
End of Module Assessment
● End of Module assessment are designed to assess all standards of the module (at least at the cluster level) with an emphasis on assessing thoroughly
those presented in the second half of the module.
● Recall, as much as possible, assessment items are designed to assess the standards while emulating PARCC Type 2 and Type 3 tasks.
● Recall, rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the assignment of grades.
Session Outcomes
What do we want participants to be able to do as a result of this
session?



Participants will draw connections between the progression documents
and the careful sequence of mathematical concepts that develop within
this module, thereby enabling participants to enact cross- grade
coherence in their classrooms and support their colleagues to do the
same.
Participants will be able to articulate how the topics and lessons promote
mastery of the focus standards and how the module addresses the major
work of the grade in order to fully implement the curriculum.
Participants will be prepared to implement the modules and to make
appropriate instructional choices to meet the needs of their students
while maintaining the balance of rigor that is built into the curriculum.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.
Session Overview
Section
Time
Overview
Introduction
28 min
Introduces Algebra II Module 3


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3
105 min
Extends students’ understanding
of exponential functions to a
domain of all real numbers and to
use their prior knowledge about


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3 Topic
Opener A
Topic A: Real
Numbers
Prepared Resources
Facilitator Preparation
families of functions to graph
exponential functions.
Topic B: Logarithms
Mid-Module
Assessment
Topic C: Exponential
and Logarithmic
Functions and Their
Graphs
Topic D: Using
Logarithms in
Modeling Situations
Topic E: Geometric
Series and Finance
End of Module
Assessment and
Conclusion
99 min
Explores developing an
understanding of the relationship
between exponentials and
logarithms and a variety of
methods for solving exponential
equations.


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3 Topic
Opener B
25 min
Allows Participants to complete a
Mid-Module Assessment and
follow up discussion.


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3 MidModule Assessment
81 min
Explores how exponential
functions are defined for all real
numbers and logarithmic
functions are defined for all
positive numbers.


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3 Topic
Opener C
87 min
Explores creating models of
contexts that involve exponential
growth and decay and connect
geometric sequences to
exponential growth and decay.


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3 Topic
Opener D
35 min
Explores how students learn about
series and sigma notation for the
first time and how they apply their
knowledge of geometric
sequences to various financial
contexts.


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3 Topic
Opener E
38 min
Revisits key points of Module 3
and allows Participants to
complete an End of Module
Assessment and follow up
discussion.


Algebra II Module 3 PPT
Facilitator Guide
Review Algebra II Module 3 and
End of Module Assessment
Session Roadmap
Section: Introduction
Time: 28 minutes
In this section, you will be introduced to Algebra II Module 3.
Materials used include:
 Algebra II Module 3 PPT
 Algebra II Module 3 Facilitators Guide
Time Slide #
3 min
1.
Slide #/ Pic of Slide
Script/ Activity directions
MATERIALS NEEDED:
• Powerpoint
• Projector
• Clicker
• Laptop
• Document camera
• Participant Binder
[Note: This session is 9 hours in length.]
Introduce presenter and talk about the session.
Mostly we will be working on the student pages. Avoid looking at the
teacher pages for the most part. I mainly want you to experience the
module from the student perspective.
5 min
2.
Start here. Then back up to opening slide.
Give participants 5 min to do the opening exercise.
We will discuss this exercise later today.
GROUP
3 min
3.
In order for us to better address your individual needs, it is helpful to
know a little bit about you collectively.
Pick one of these categories that you most identify with. As we go
through these, feel free to look around the room and identify other folks
in your same role that you may want to exchange ideas with over lunch or
at breaks.
By a show of hands who in the room is a classroom teacher?
Math trainer?
Principal or school-level leader
District-level leader?
And who among you feel like none of these categories really fit for you.
(Perhaps ask a few of these folks what their role is).
Regardless of your role, what you all have in common is the need to
understand this curriculum well enough to make good decisions about
implementing it. A good part of that will happen through experiencing
pieces of this curriculum and then hearing the commentary that comes
from the classroom teachers and others in the group.
2 min
4.
We have three main objectives for this morning’s work. Our main task
will be experiencing lessons and assessments. As a secondary objective,
you should walk away from the study of Module 3 being able to articulate
how these lessons promote mastery of the standards and how they
address the major work of the grade. Lastly, you should be able to get a
sense for the coherent connections to the content of earlier grade levels.
2 min
5.
Here is our agenda for the day.
We will spend most of our time on G11 M3. As we go through the
module, I will talk about foundational skills developed in prior grades.
We will discuss some fluency drills and other scaffolds that can be used to
address possible gaps in content knowledge.
We will also discuss how this develops skills needed for grade 12
(Precalculus).
3 min
6.
(Go through the bullets to give an overview of the progression or flow of
each topic and the module as a whole.)
MMA after topic B. 33 lessons. 45 instructional days
5 min
7.
Supplies: Teacher pages 1 – 10
Display and have participants read the Module overview, scan the
standards and the new terminology
8 min
8.
(Review the bullet points with participants to remind them of the
background students are coming in to this module with.)
We are extending our understanding of exponential functions from
integers to all real numbers.
Students spent a good bit of time in Algebra I comparing linear and
exponential growth and connecting that to arithmetic and geometric
sequences.
Section: Topic A: Real Numbers
Time: 105 minutes
In this section, you will explore extending students’ understanding
of exponential functions to a domain of all real numbers and to use
their prior knowledge about families of functions to graph
exponential functions.
Materials used include:
 Algebra II Module 3 PPT
 Algebra II Module 3 Facilitators Guide
Time Slide #
0 min
9.
Slide #/ Pic of Slide
Script/ Activity directions
GROUP
5 min
10.
Supplies: Teacher p. 11-13
Read the Topic A Opener
10
min
11.
Supplies: S1 and 2, toilet paper
In Algebra I, students explored the power of exponential growth
(including the paper folding problem) and studied exponential functions
but only with a domain of the integers. This is largely a review.
As we discussed in module 1, the idea of “simplifying” is not emphasized
here. It is the idea of rewriting an expression in a variety of ways.
Which form we chose depends on what our goal is.
5 min
12.
It depends on my purpose. If I am trying to evaluate this expression for
several values of x, I would probably prefer the first. If I am about to
find the derivative, I would prefer the second.
If I am going to evaluate the expression or set it equal to a number and
solve an equation, these tasks will be much easier if the expression is
reduced as much as possible. In Calculus, if this is the first derivative,
and I am trying to find the second derivative, it will be much easier
reduced.
10
min
13.
Supplies: student pages 10 and 11
If you ask students to explain scientific notation, they will probably talk
about moving decimals left and right and counting to find a positive or
negative exponent. I find they rarely connect that to the idea of
multiplying by powers of 10 especially when negative exponents are
involved.
You may want to do some calculator work here. Show them the 10^x
button. On most scientific calculators it is above this weird button “log.”
Why would that be?
Look at exercise 7 and 8. Work a few from Example 2 and/or Exercise 9
– 11.
2 min
14.
Students began exploring properties of exponents in 8th grade and
continued in Algebra I. In Algebra II, the standards call for students to
extend their understanding of exponents to include rational exponents.
3 min
15. 15.
In Algebra I, students graphed exponential functions. They only worked
with a domain of integers, but often they would draw a curve through
the points. This seems to imply that the domain is the set of all real
numbers.
3 min
16. 16.
Estimate 2^(1/2) from the graph. Seems to approximately equal 1.4
…hmm √2 ≈ 1.4
1
1
1
1
1
1
1
1
1
1
2𝑛 = √2 because (2𝑛 )^𝑛 = (2𝑛 ) (2𝑛 ) (2𝑛 ) … (2𝑛 ) = 2𝑛+𝑛+𝑛+⋯𝑛 = 2
𝑛
2 min
17.
Talk through lesson summary. Note the definition of the nth root of a
number and the principal nth root of a number.
10
min
18.
Supplies: S.24 – 27
Remember that we are not asking students to “simplify” which is vague
and unclear.
Switch over to the doc cam and work example 3, 4, and exercise 5 (a).
Point out the directions and that there may be more than one approach
(show approach from teacher pages for example 4 and then share other
approaches from participants)
10
min
19.
Supplies: S.30
How do we go about raising 2 to the power of √2? How do we make
sense of this within the context of an exponent?
Work through Exercise 1.
5 min
20.
As we raise 2 to numbers approaching √2, we see that we have to go
further back in the decimal to detect a change. Once a decimal place
“settles” on a value, it no longer changes as the exponent gets closer to
√2.
So our approximation gets better the larger we make the index of the
radical, but we could keep doing this to infinity. So 2^√2 is the limit as n
approaches infinity. This is the same way a calculator or computer
approximates a value.
Compare that to 2^√2 on the calculator.
5 min
21.
Let’s look at another approach. We want to “trap” the value of 2^√2
using smaller and smaller intervals.
How far could we go? We could continue forever making our interval
smaller and smaller?
What happens to our solution as we shrink the interval? Our solution
gets more accurate. More decimal places “settle” and we have to go
further back to find a decimal place that is changing.
30
min
22.
Supplies: S.34 – 41 and T.93
Euler’s number is hugely important in mathematics but is rather
mysterious to the students. In this lesson, they define it through an
exploration. It is important to emphasize that we have “discovered” this
number using one specific example. The number e is remarkable but
not magic.
It is important in population growth, finance, and other instances. They
will see the number e again in precalculus and calculus.
I am going to model this lesson as if you were the students.
5 min
23.
5 min. Total time elapsed: 127 minute + 8 minutes
discussion/questions = 135 minutes or 2 hours 15 minutes
(Go through each point listed.)
Students have built on and expanded their understanding of exponents
which began in 8th grade and exponential functions which began in
Algebra I. This leads us to a key topic in Algebra II – logarithms.
Section: Topic B: Logarithms
Time: 99 minutes
In this section, you will explore developing an understanding of the Materials used include:
relationship between exponentials and logarithms and a variety of
 Algebra II Module 3 PPT
methods for solving exponential equations.
 Algebra II Module 3 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
0 min
24.
5 min
25.
5 minutes
Supplies: Teacher p. 102-104
Read topic opener for topic B
3 min
26.
Students first solve exponential functions algebraically but quickly realize
that this technique limits what type of exponential equations we are able to
solve. Not all exponential equations can be rewritten using similar bases.
And most exponential equations do not have a rational solution. Now we
are dealing with an equation whose solution is irrational.
Students then take a numeric approach similar to what we saw in topic A.
They find the solution by using a “squeeze” method that eventually traps the
solution.
5 min
27.
As the class hones in on the correct solution, work can be divided to speed
up the process.
We can find the solution to any level of precision desired. If I want a
solution accurate to 4 decimal places, I need to continue until the first four
decimal places have “settled” and agree.
Students solve a similar problem but within the context of an E. Coli bacteria
colony which doubles in size every 30 minutes.
10 min
28.
Supplies: S.49 – 51
Have participants work the opening exercise. This is how we open the
lesson for students. No instructions, no explanations.
The lesson opens with a new function called a “whatpower” function.
Without giving any explanation, ask students to evaluate each expression.
Allow students to struggle through interpreting what the function is asking
them to do. Put them in pairs if necessary. Even if they don’t finish all of
them, make sure everyone gets to the last question.
Students are making sense of the definition of a logarithm without being
confronted with the strange terminology. It is a more intuitive approach.
Before introducing the log notation, students already have an idea of how
this function works and that there are restrictions on the base as well as on
the input.
3 min
29.
Then the logarithm function is introduced. As we did with the trigonometric
function, be diligent about using parentheses throughout the study of
logarithms to emphasize that it is a function. log of a number  log(100)
This Frayer model is a great way to summarize the discussion about
logarithms.
10 min
30.
10 min
Supplies: personal board, dry erase markers, felt
Factoring is a skill that students need to develop fluency with. It is a great
example of a skill for which a rapid white board exchange is a fitting fluency
exercise.
How to conduct a white board exchange:
All students will need a personal white board, white board marker, and a
means of erasing their work. An economical recommendation is to place
card stock inside sheet protectors to use as the personal white boards, and
to cut sheets of felt into small squares to use as erasers. You have these
materials at your tables today.
It is best to prepare the questions in a way that allows you to reveal them to
the class one at a time. A flip chart, or Powerpoint presentation can be used,
or one can write the problems on the board and either cover some with
paper or simply write only one on the board at a time.
Prepare 10-15 problems that progress in difficulty. The best way to get the
feel is for us to do one ourselves. I’ll reveal the problem, you work it as fast
as you can and still do accurate work and then hold it up for me to see.
(Reveal the first problem in the list and announce, “Go”. Give immediate
feedback to each participant, pointing and/or making eye contact with the
participant and responding with an affirmation for correct work such as,
“Good job!”, “Yes!”, or “Correct!”, or guidance for incorrect work such as
“Look again,” “Try again,” “Check your work,” etc. Do several to demonstrate
the progression of problems.)
If many students have struggled to get the answer correct, go through the
solution of that problem as a class before moving on to the next problem in
the sequence. Fluency in the skill has been established when the class is
able to go through each problem in quick succession without pausing to go
through the solution of each problem individually. If only one or two
students have not been able to get a given problem correct when the rest of
the students are finished, it is appropriate to move the class forward to the
next problem without further delay; in this case find a time to provide
remediation to that student before the next fluency exercise on this skill is
given.
5 min
31.
Look back at the opening exercise. Discuss responses. How did you come
up with your answer?
5 min
32.
Lesson 9 motivates the use of logarithms by exploring the idea of assigning
digits as unique identifiers.
There are approximately 317 million people in the U.S.
Log(100,000,000)=8
Log(1,000,000,000)=9
9 digits will be enough unless the population reaches 1 billion.
10 min
33.
Supplies: S. 60 – 64
Scan the opening exercise.
Students work with the common logarithm in this lesson to discover some
properties of base 10 logs that will later lead to properties of logs of any
base b (>0 and not equal to 1).
Work exercises 7 – 14.
Look at lesson summary. These properties will later be expanded to include
logs of any base.
10 min
34.
Supplies: S. 68-71
Work Exercises 1 – 5 and part(s) of example 1 and example 2.
Why is this “the most important property” of logarithms? Gives us a way of
evaluating logarithms using only a few known values. Many of the other
properties of logs can be derived from this one.
3 min
35.
Students continue to explore properties of logarithms using the definition of
a logarithm as well as the properties developed in lessons 10 and 11.
Students use properties 3 and 6 to solve exponential equations.
5 min
36.
We could evaluate a base 2 log using a squeeze process as we have in other
instances in this module. That is a long and tedious process.
In this lesson students use the change of base formula to evaluate
logarithms and solve exponential equations.
10 min
37.
Supplies: S. 91 – 93
Students should look for ways to use the properties of logarithms to make
the process of solving the equation easier.
Anytime there is a restricted domain, we must check for extraneous
solutions (remind students of past experiences with radical and rational
equations).
Work Exer 4, 5, 8. For 5 and 8, show two approaches.
10 min
38.
Supplies: teacher pages 214, 223,224
Takes students through the historical development of logarithms, log tables,
and how they made computations done without the aid of a computer or
calculator much easier.
Work through examples 1 – 3 with the participants (not in student pages)
and exit ticket.
5 min
39.
Total time elapsed: 240 minutes + 15 minute break = 250 minutes or 4
hours 10 minutes
(Go through each point listed.)
Students should have a solid understanding of the logarithm as a function.
Hopefully we can avoid those algebraic mistakes we tend to see with logs
(show a few examples).
Students have seen that logarithms are useful particularly in scientific
applications when working with very large or very small numbers.
Section: Mid-Module Assessment
Time: 25 minutes
In this section, you will allow Participants to complete a MidModule Assessment and follow up discussion.
Materials used include:
 Algebra II Module 3 PPT
 Algebra II Module 3 Facilitators Guide
Time
0 min
Slide # Slide #/ Pic of Slide
40.
Script/ Activity directions
GROUP
15 min
41.
Supplies: Teacher p. 225 – 232
Have participants locate the assessment. Give them approximately 15 min
to take the assessment with their partner. After 10 minutes have passed
give a verbal warning for them to scan any remaining questions that they
have not yet attempted. If everyone finishes early, stop this part and start
the next portion of this session.
10 min
42.
Total time elapsed: 275 min or 4 hrs 35 min.
Again, work with a partner to examine your work against the rubric and
exemplar. If you have any questions or concerns, jot them down on a postit note and we will address those before we move on.
After 10 minutes or so have passed, call the group together and address
any questions or concerns that participants noted on their post-it notes.
Section: Topic C: Exponential and Logarithmic Functions
and Their Graphs
Time: 81 minutes
In this section, you will explore how exponential functions are
defined for all real numbers and logarithmic functions are defined
for all positive numbers.
Materials used include:
 Algebra II Module 3 PPT
 Algebra II Module 3 Facilitators Guide
Time
Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
0 min
43.
2 min
44.
Supplies: Teacher p. 246 – 248
Go through each bullet point.
Read through the Topic C Opener.
3 min
45.
We tend to think that most numbers are rational and that there are a few
examples of numbers that are irrational. In fact, the opposite is true. There
are far more irrational numbers than rational. Rational numbers are
countably infinite while irrational numbers are an uncountable infinite.
3 min
46.
Students continue to explore the notion of “trapping” an irrational solution
as a lead-in to graphing logarithmic functions.
2 min
47.
Students are divided into teams. Each team is either a “10 team”, a “2
team”, or a “5 team.”
2 min
48.
By comparing results as a class, students see that these graphs have many
features in common. In this way, they identify the key features of the graph
of a logarithmic function.
8 min
49.
Supplies: S.109
Students remain in their teams and explore the graph of 1/b. The graphs
are reflections across the x-axis of the original graphs. Why would that be?
Next, students explore the graph of y =𝑙𝑜𝑔𝑏 (𝑏𝑥). From past experience, we
know that b should be a horizontal stretch or compress.
Look at exercise 3. Graph y = log(x) and y = log(10x) together. What effect
did the 10 have on the graph?
3 min
50.
Supplies: S.117
The goal here is to realize that there is a connection between the two
graphs and that they have “opposite” properties in many ways. We are
leading students to the idea of inverse functions. However, the term
inverse is not used in this lesson.
Look at exercise 1.
3 min
51.
Inverse functions will be covered in much more detail in precalculus (the
plus standards). At this point, students have not been introduced to
inverse functions but it seems logical to discuss them within the context of
exponential and logarithmic functions.
In addition to exponential and logarithmic, students also look at some
examples using linear and cubic functions (they graphed these in algebra I).
13 min
52.
Supplies: graphing calculators, S.124,125
Understanding the inverse relationship is a plus standard, but necessary to
solve exponential equations. Students will only need to understand this
standard to the degree that it is needed to rewrite between logarithmic and
exponential form within the context of solving equations. It will be covered
in more detail in Precalculus.
Work exercises 3, 5, and 7.
Note that they find the equation of the inverse but then use technology to
produce the graphs of the function and its inverse and use the graph as a
means of checking the equation.
15 min
53.
Supplies: S.130-136
Students see that properties of exponents and logarithms can produce
some surprising results. Unlike graphs of functions studied previously,
sometimes two different transformations can produce the same graph.
Using the properties and noting the structure of the function can make
graphing exponential and logarithmic functions easier.
Work through the exploratory challenge and examples 1 and 2.
5 min
54.
Supplies: T324
Look at lesson summary points. How are the general forms helpful for
graphing? Work through the exit ticket from the lesson.
5 min
55.
Supply: S.141, 142
The graphs of y = 𝑙𝑜𝑔2 𝑥 and y = 𝑙𝑜𝑔10 𝑥 are already provided. Ask students
to identify which graph is y = 𝑙𝑜𝑔2 𝑥 and which is y = 𝑙𝑜𝑔10 𝑥.
Students then graph y = ln(x) on the same axes and compare it with the
other two graphs. It is difficult to explain to students at this point why we
would care about the natural log, but it is of great importance in calculus
since ∫ 1/𝑥 𝑑𝑥 = ln(𝑥).
Students usually struggle with the natural log function. Continue to
emphasize that since 2 < e < 3, y = ln(x) looks very similar to y = 𝑙𝑜𝑔2 𝑥 and
y = 𝑙𝑜𝑔3 𝑥.
5 min
56.
Supply: S.141, 142
Look at example 2. We can rewrite 𝑙𝑜𝑔6 (𝑥) as any base we choose. Let’s
rewrite it as a natural log.
𝑙𝑛(𝑥)
• 𝑔(𝑥) = ln(6) − 2
•
1
ln(6)
≈ 0.558
We can take the graph of y = ln(x) and scale it vertically by an approximate
factor of 0.558 and then translate the graph down 2 units. Compare that to
the graph of 𝑔(𝑥) = 𝑙𝑜𝑔6 (𝑥) − 2. What would be the advantage of
rewriting it in this way?
10 min
57.
Supply: Teacher p. 357
These are the primary functions they have used in modeling. Also used
cubic and root functions.
Students explore this question. This lesson is primarily a review of
modeling data. Leading in to topic D where we are modeling with
logarithmic functions.
Work the exit ticket.
2 min
58.
Total time elapsed: 356 min + 9 minutes questions/discussion = 365
minutes or 6 hrs 5 min.
Go through each bullet point and ask participants to add any. How are
properties of logarithms and exponents useful when graphing? Why is it
important for students to understand this inverse relationship between
logarithmic functions and exponential functions?
Section: Topic D: Using Logarithms in Modeling Situations Time: 87 minutes
In this section, you will explore creating models of contexts that
involve exponential growth and decay and connect geometric
sequences to exponential growth and decay.
Time
0 min
Slide # Slide #/ Pic of Slide
59.
Materials used include:
 Algebra II Module 3 PPT
 Algebra II Module 3 Facilitators Guide
Script/ Activity directions
•
GROUP
5 min
60.
Supplies: teacher p. 361 – 363
Go through each bullet point.
Read Topic Opener for Topic D.
25 min
61.
Supplies: Teacher p. 364-380, graphing calculators, beans or M&Ms, plastic
cups, paper plates.
Lead participants through a lesson study. Have them work through the
activity and read the teacher pages making notes of how they would
implement the lesson in their class. Discuss as a group what
implementation would look like. Timing? Two days for the lesson?
Instructional decisions.
15 min
62.
Supplies: S. 162,163
Think back through the ways in which we have solve exponential
equations. Relating the bases, using a “squeeze” method, using properties
of logarithms, applying the change of base formula.
This is the first time we have solved exponential equations within a
modeling context.
Work this problem two different ways.
Work through parts of exercise 2 and 3.
3 min
63.
Let’s go through students’ experiences in Algebra I.
In Module 3 of Algebra I students learned how to write both recursive and
explicit formulas for a variety of sequences.
3 min
64.
They also learned how to recognize arithmetic and geometric sequences
and learned that arithmetic sequences were characterized by a constant
rate of change and geometric sequences were characterized by a constant
growth factor.
3 min
65.
Students learned about linear and exponential growth. One of the major
themes of this module was that an increasing geometric function will
always exceed an increasing linear function at some point.
13 min
66.
Supply: S. 173,174
Here they wrote both a linear and exponential model and then compared
them to the actual census figures to determine the validity of the models.
Some of this lesson will be review, but the problems are increasing in
complexity and adding new components such as the number e.
Work through parts of exercises 2 and 3.
2 min
67.
Students used the first two formulas in Algebra I. We are adding the third
formula and also the ability to solve for t using logarithms.
3 min
68.
We have increased the complexity of the task and added additional tools
from Algebra I.
5 min
69.
Students explored coffee cooling under different conditions using Wolfram
Alpha exploration and then graphs and transformations.
In lesson 28, we explore this law again.
Show demonstration on Wolfram Alpha.
8 min
70.
Supply: S.198-200
You will need to review through the parameters with the students.
Work through math modeling exercises 1 and 2 as time permits.
2 min
71.
Total elapsed time: 452 min + 15 min break + 8 min questions/discussions
= 475 min or 7 hrs 55 min.
Review through each bullet point.
Section: Topic E: Geometric Series and Finance
Time: 35 minutes
In this section, you will explore how students learn about series and Materials used include:
sigma notation for the first time and how they apply their
 Algebra II Module 3 PPT
knowledge of geometric sequences to various financial contexts.
 Algebra II Module 3 Facilitators Guide
Time
Slide # Slide #/ Pic of Slide
0 min
72.
5 min
73.
Script/ Activity directions
Supplies: Teacher p. 469 – 470
Read Topic Opener for Topic E.
GROUP
3 min
74.
Students continue to work with sequences in this part of the module,
primarily geometric sequences but arithmetic sequences are reviewed in
the problem sets. In this lesson, students develop a formula for the sum of
a finite geometric series. They also use sigma notation for the first time.
3 min
75.
You could have students verify their sums using a spreadsheet or a
graphing calculator.
5 min
76.
Illustrate this problem using a spreadsheet.
3 min
77.
In this open-ended exploration, students determine the monthly payment
on a car of their choosing. They also consider tax, title, insurance, and
other costs that should be considered before purchasing a car.
8 min
78.
Supplies: teacher page 510 – 512
Illustrate with spreadsheet. Look through the teacher pages of the
challenge. How will you incorporate technology?
3 min
79.
Students complete this open-ended modeling task with a home of their
choosing.
3 min
80.
This culminating lesson of the module takes students through the full
modeling cycle and incorporates all of the formulas developed through
topic E.
2 min
81.
Go through each bullet point.
Section: End of Module Assessment and Conclusion
Time: 38 minutes
In this section, you will revisit key points of Module 3 and allow
Participants to complete an End of Module Assessment and follow
up discussion.
Materials used include:
 Algebra II Module 3 PPT
 Algebra II Module 3 Facilitators Guide
Time
Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
5 min
82.
Total elapsed time: 515 min. or 8 hrs 35 min.
Leave this slide blank.
Ask participants to list key points or ideas from module 3.
Take a moment now to re-read the standards that this module covers… Can
you think back to moments in the lessons that get students to arrive at
those understandings? What things stand out to you now that did not
stand out early on?
0 min
83.
15 min
84.
Supplies:
Have participants locate the assessment. Give them approximately 15 min
to take the assessment with their partner. After 10 minutes have passed
give a verbal warning for them to scan any remaining questions that they
have not yet attempted. If everyone finishes early, stop this part and start
the next portion of this session.
10 min
85.
Again, work with a partner to examine your work against the rubric and
exemplar. If you have any questions or concerns, jot them down on a postit note and we will address those before we move on.
After 10 minutes or so have passed, call the group together and address
any questions or concerns that participants noted on their post-it notes.
3 min
86.
(Review each key point one at a time.)
5 min
87.
Total time elapsed: 548 minutes or 9 hours and 8 minutes
Take a few minutes to reflect on this session. You can jot your thoughts on
your copy of the PowerPoint. What are your biggest takeaways? (pause
while participants reflect then click to advance to the next question). Now,
consider specifically how you can support successful implementation of
these materials at your schools given your role as a teacher, school leader,
administrator or other representative.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Active learning
Turn and talk
Turnkey Materials Provided
●
●
Algebra II Module 3 PPT
Algebra II Module 3 Facilitator’s Guide
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Additional Suggested Resources
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How to Implement A Story of Functions
A Story of Functions Year Long Curriculum Overview
A Story of Functions CCLS Checklist